我可以開公司嗎?
No matter what status you are, an F-1, H-1, using EAD, etc, the short answer is "Yes".
However, you can not work, even for the company you established, unless you are in possession of an H-1 sponsored by your company, or a valid EAD. You may own that business, you may hire someone to run the business for you, you may earn profit from your business, but you just can not work for it, such activity constitutes unauthorized working.
To start a business from scratch? Google "LLC".